2. (4pts.) Consider the circuit below. The capacitor is initially uncharged and then the switch is closed at t = I]. In this problem, we will see how the limiting behavior of capacitors we leamed (will learn) in class. is applicable to any circuit. for any t b I]. This is a differential equation for Q{t}, but you do not need any previous knowledge about differential equations to get to this relation or to solve the rest of this problem Hint: Write the current through the capacitor in terms of Q{t}. (b) Find the value of R that makes the expression 0532 4/:'20 t = 1 Qt ) R1 + 82 i e ) satisfy the equation you found in part (a) by directly substituting this expression in. (c) Use the solution from (b) to nd the current through each of the resistors and the current through the wires connecting the capacitor. all as a function of time and in terms of C, R1. R2, and 8, or a subset of these quantities. (d) Plot (by hand) all the currents found in part (c). Make sure to specify their values in the limits t > {J and att > 00. This limits are equivalent to t K T and t >> T, respectively, where T 2 RC. (e) Using the methods learned in class to deal with the limiting behavior of capacitors, nd the current through each of the resistors right before and a long time after the switch is closed. Check that these values are the same as what you obtained in part (d) when taking the limits t > I] and t lr 00. (f) Using the results and method from part (e), show that the nal charge of the capacitor agrees with the expression given in part (b)